Please answer all questions and show work for all questions. Answer key is attached, the work shown is the most most important part. Please write neatly, write explanations and step by step breakdowns for every problem, and write on paper do not type it. All the problems are shown in the pictures and the last picture contains some formulas that are needed for the word problems. There is no missing content all is in the pictures, please do not report. Thank you!

2. Find the indicated indefinite integrals. a) (15e3x + e2)dx b) 5 6 -3/2 + 6x - 4 7-5 + . dx x c ) ( 2 x + 3 ) 2 dx d ) dx Vx 4x3 a) 5e3x +e2x + C b) -10x-1/2 + = x-4+31n/x|+C c) 3 + 6x2 +9x + C x2 -3 d ) ? + 4x3/2 -8x1/2+ C e) x1/3 + 2x* - In |x| + C 3. Find the solution to each of the given initial value problems. a) f'(x) = 2x + 3 and f(1) =2 b)h'(t) = 6e2t and h(0) =e c) f'(x) =5 -= and f(-1) = 0 a) f(x) = x2+ 3x -2 b) h(t) = 302t + e -3 c)f(x) =5x -In |x|+54. Compute the following indefinite integrals. a ) 3 6x2+ 2 (2x + 7)3 ax b ) 8e- 4x+ 2 dx c) 15x*(x5 + 3)-3 dx d ) dx x3+x+5 e ) xV2x - 1dx X f) (x -5)3 dx 3 3 a b) - 2e-4x+2 + C C ) - d) 2 In|x3 + x + 51+ C 4(2x + 7)2 + C 2(x5 + 3)2 + C -1 5 e) 10 (2x - 1)5/2 += (2x -1)3/2+ C f) x - 5 2(x -5)2 + C7. Evaluate each definite integral. Make sure to show all of your work. 1 me2 a ) / (x2 - 4x) dx b) -dx c ) 9x1/2dx d ) xex? (In t) 2 dx e) Jo e t -at f ) xv 4 - xdx -1 X o a) - 3 b) 0.69 c) 48 d) 0.86 e) 2.33 f) 8.53 8. Determine the area of the specified region. Draw a graph of the functions and shade the appropriate region. You should compute the integrals by hand, but you may use your calculator to check your answer. a) Area under f (x) = between x = -1 and x = 4 x4+1 x 2 b) Area under f (x) = - e-1/* between x = 1/2 and x = 2 c) Region enclosed by y = 2x2 and y = x2 + 3 a) 1.214 b) 0.471 c) 6.9289. Solve the following applied problems. Write your answers in complete sentences and include appropriate units. a) You are applying for a car loan. Two banks are offering you the following plans: Bank A: 13.2% interest compounded semiannually Bank B: 12.9% interest compounded monthly Which loan would be a better deal? Bank A has an APY of 13.64%, and Bank B has an APY of 13.69%. Bank A is a better deal for a loan. b) Find the APY for money invested at 9.6% compounded semiannually, 9.5% compounded weekly, and 9% compounded continuously. Which would be the best investment? The APY in each case is 9.83%, 9.96%, and 9.41%. The best deal for an investment is 9.5% compounded weekly. New parents deposit $10,000 in a certificate of deposit on the day their daughter is born. If the account pays 4.5% compounded continuously, how much money will be in the account the day that the daughter turns 18 years old? There will be $22,479 in the account. d) What is the annual interest rate if an investment of $5,000 compounded monthly doubles it's value in 6 years? The annual interest rate is 11.6%. e) A family has a $500,000 mortgage at 6.1% compounded monthly for 30 years. Find the monthly payment. How much interest would they have paid by the time they pay off their loan? The monthly payment is $3,029.97. The total interest they will pay is $590,789.20. Parents have set up a sinking fund in order to have $120,000 in 15 years for their children's college education. How much should be paid semiannually into an account paying 4.8% compounded semiannually? The semiannual payments are $2,777.g) E-loan recently offered 36-month auto loans at 0.21% monthly rate to applicants with fair credit ratings. If you have a fair credit rating and can afford monthly payments of $350, how much can you borrow from E-loan? What is the total interest that you will pay for a loan of this amount? You can borrow up to $12.123.25, and you will pay $477 in interest for this loan. h) If you buy a computer directly from the manufacturer for $2,500 and agree to repay it in 48 equal installments at 1.25% interest per month on the unpaid balance, how much are your monthly payments? How much total interest will be paid? Your monthly payments are $69.58. You will pay a total of $839.84 in interest. i) A family is thinking about buying a house costing $350,000. They will pay 20% down, and the rest is to be amortized over 30 years in equal monthly payments. If their mortgage has an annual interest rate of 4.4% compounded monthly, what will be the monthly payments? How much total interest will be paid over 30 years? The monthly payments will be $1,402. They will pay a total of $224,720 in interest. j) Rafael makes his first $1,000 deposit into an IRA earning 4.1% interest compounded annually on his 24th birthday and his last $1,000 deposit on his 35th birthday (12 equal deposits in all). With no additional deposits, the money in the IRA continues to earn 4.1% interest compounded annually until Rafael retires on his 65th birthday. How much is in the IRA when he retires? How much total interest did he earn? Rafael will have $50,448.92 when he retires and will have earned $38,448.92 in interest. k) A company that services vending machines considers its income as a continuous stream with an annual rate of flow given by f(t) = 120e-0.4 in thousands of dollars. Find the income from this stream over the next 10 years. What is the income produced in the last year only? (Hint: you are not investing the income in this problem) The income over 10 years is $294,505. The income during only the 9th year is $2,702. 1) A continuous income stream has an annual rate of flow of f(t) = 12,000e0.04t (in dollars). Find the future value if the income is compounded continuously at 6% for the next 8 years. The future value after 8 years is $143,368.m) Starting at age 25, you deposit $2,000 each year into an IRA. Treat the yearly deposits into the account as a continuous stream ofincome. lfmoney in the account earns 5% interest compounded continuously, how much will be in the account when you retire at age 65 (40 years later)? How much ofthe final amount is interest? There will be $255,562 in the account at age 65. The interest is $175,562. 11) A small oil company considers the continuous pumping of oil from a well as a continuous income stream with an annual rate of ow off(t) = 60094\" in thousands of dollars. Find the future value ofthe income stream if it's compounded continuously at 10% over the next 10 years. What interest will this investment pay? The future value after 10 years is $5,165,893. The total interest is $2,571,899. The following nancial formulas will be given to you on the exam. No other formulas will be provided A=P+1 I=Prt A=P(1+rt) mt A=P(1+E) =P(1+i)\" A=Pert r m APY=(1+m) 1 0R APY=er1 (1+ 1')\" 1 FV =PMT+ 1(1+i)'\" i PV=PMT T = C! Q [a fit) t T FV = 9\"] f(t)e'\"dt U